Hostname: page-component-745bb68f8f-l4dxg Total loading time: 0 Render date: 2025-01-31T10:55:40.176Z Has data issue: false hasContentIssue false
  • English
  • Français

Construction of reduced-order models for fluid flows using deep feedforward neural networks

Published online by Cambridge University Press:  14 June 2019

Hugo F. S. Lui  and
William R. Wolf Open the ORCID record for William R. Wolf [Opens in a new window]
Hugo F. S. Lui
Affiliation:
School of Mechanical Engineering, Universidade Estadual de Campinas, Campinas, SP13083-860, Brazil
William R. Wolf*
Affiliation:
School of Mechanical Engineering, Universidade Estadual de Campinas, Campinas, SP13083-860, Brazil
*
†Email address for correspondence: wolf@fem.unicamp.br
Get access Rights & Permissions [Opens in a new window]

Abstract

We present a numerical methodology for construction of reduced-order models (ROMs) of fluid flows through the combination of flow modal decomposition and regression analysis. Spectral proper orthogonal decomposition is applied to reduce the dimensionality of the model and, at the same time, filter the proper orthogonal decomposition temporal modes. The regression step is performed by a deep feedforward neural network (DNN), and the current framework is implemented in a context similar to the sparse identification of nonlinear dynamics algorithm. A discussion on the optimization of the DNN hyperparameters is provided for obtaining the best ROMs and an assessment of these models is presented for a canonical nonlinear oscillator and the compressible flow past a cylinder. Then the method is tested on the reconstruction of a turbulent flow computed by a large eddy simulation of a plunging airfoil under dynamic stall. The reduced-order model is able to capture the dynamics of the leading edge stall vortex and the subsequent trailing edge vortex. For the cases analysed, the numerical framework allows the prediction of the flow field beyond the training window using larger time increments than those employed by the full-order model. We also demonstrate the robustness of the current ROMs constructed via DNNs through a comparison with sparse regression. The DNN approach is able to learn transient features of the flow and presents more accurate and stable long-term predictions compared to sparse regression.

JFM classification

Mathematical Foundations: Computational methods Nonlinear Dynamical Systems: Low-dimensional models
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 872 , 10 August 2019 , pp. 963 - 994
Copyright
© 2019 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Abadi, M. et al. 2016 TensorFlow: Large-scale machine learning on heterogeneous distributed systems. arXiv:1603.04467.Google Scholar
Akaike, H. 1973 Information Theory and an Extension of the Maximum Likelihood Principle, pp. 199–213. Springer.Google Scholar
Aris, R. 1989 Vectors, Tensors, and the Basic Equations of Fluid Mechanics. Dover.Google Scholar
Baik, Y. S., Rausch, J., Bernal, L. & Ol, M. V. 2009 Experimental investigation of pitching and plunging airfoils at Reynolds number between 1 × 104 and 6 × 104 . In AIAA Paper 2009-4030.Google Scholar
Balajewicz, M., Tezaur, I. & Dowell, E. 2016 Minimal subspace rotation on the Stiefel manifold for stabilization and enhancement of projection-based reduced order models for the compressible Navier–Stokes equations. J. Comput. Phys. 321, 224–241.Google Scholar
Beam, R. M. & Warming, R. F. 1978 An implicit factored scheme for the compressible Navier–Stokes equations. AIAA J. 16 (4), 393–402.Google Scholar
Bengio, Y. 2009 Learning deep architectures for AI. Found. Trends Mach. Learn. 2 (1), 1–127.Google Scholar
Bergstra, J. & Bengio, Y. 2012 Random search for hyper-parameter optimization. J. Mach. Learn. Res. 13, 281–305.Google Scholar
Brochu, E., Cora, V. M. & Freitas, N.2010 A tutorial on Bayesian optimization of expensive cost functions, with application to active user modeling and hierarchical reinforcement learning. arXiv:1012.2599.Google Scholar
Brunton, S. L. & Noack, B. R. 2015 Closed-loop turbulence control: progress and challenges. Appl. Mech. Rev. 67, 050801.Google Scholar
Brunton, S. L., Proctor, J. L. & Kutz, N. J. 2016 Discovering governing equations from data by sparse identification of nonlinear dynamical systems. Proc. Natl Acad. Sci. USA 113 (15), 3932–3937.Google Scholar
Carlberg, K., Bou-Mosleh, C. & Farhat, C. 2011 Efficient non-linear model reduction via a least-squares Petrov–Galerkin projection and compressive tensor approximations. Intl J. Numer. Meth. Engng 86 (2), 155–181.Google Scholar
Cazemier, W., Verstappen, R. W. C. P. & Veldman, A. E. P. 1998 Proper orthogonal decomposition and low-dimensional models for driven cavity flows. Phys. Fluids 10 (7), 1685–1699.Google Scholar
Chaturantabut, S. & Sorensen, D. 2010 Nonlinear model reduction via discrete empirical interpolation. SIAM J. Sci. Comput. 32, 2737–2764.Google Scholar
Clainche, S. & Vega, J. M. 2017 Higher order dynamic mode decomposition to identify and extrapolate flow patterns. Phys. Fluids 29, 084102.Google Scholar
Clevert, D., Unterthiner, T. & Hochreiter, S.2015 Fast and accurate deep network learning by exponential linear units (ELUs). arXiv:1511.07289.Google Scholar
D’Lelis, B. O. R., Wolf, W. R., Yeh, C. & Taira, K.2019 High-fidelity simulation and flow control of a plunging airfoil under deep dynamic stall. AIAA Paper 2019-1395.Google Scholar
Goodfellow, I., Bengio, Y. & Courville, A. 2016 Deep Learning. MIT Press.Google Scholar
Green, M. A., Rowley, C. W. & Haller, G. 2007 Detecting vortex formation and shedding in cylinder wakes using Lagrangian coherent structures. J. Fluid Mech. 572, 111–120.Google Scholar
Haller, G. 2015 Lagrangian coherent structures. Annu. Rev. Fluid Mech. 47, 137–162.Google Scholar
Juang, J. N. & Pappa, R. S. 1985 An eigensystem realization algorithm for modal parameter identification and model reduction. J. Guid. Control Dyn. 8, 620–627.Google Scholar
Kennedy, C. A., Carpenter, M. H. & Lewis, R. M.1999 Low-storage, explicit Runge–Kutta schemes for the compressible Navier–Stokes equations. ICASE Report No. 99-22.Google Scholar
Kingma, D. P. & Ba, J. 2014 Adam: a method for stochastic optimization. In Proceedings of the 3rd International Conference on Learning Representations (ICLR). arXiv:1412.6980.Google Scholar
Krizhevsky, A.2009 Learning multiple layers of features from tiny images. Tech. Rep.Google Scholar
Kutz, J. 2017 Deep learning in fluid dynamics. J. Fluid Mech. 814, 1–4.Google Scholar
Lele, S. K. 1992 Compact finite difference schemes with spectral-like resolution. J. Comput. Phys. 103 (1), 16–42.Google Scholar
Ling, J., Kurzawski, A. & Templeton, J. 2016 Reynolds averaged turbulence modelling using deep neural networks with embedded invariance. J. Fluid Mech. 807, 155–166.Google Scholar
Ling, J. & Templeton, J. 2015 Evaluation of machine learning algorithms for prediction of regions of high Reynolds averaged Navier Stokes uncertainty. Phys. Fluids 27, 085103.Google Scholar
Luhar, M., Sharma, A. S. & McKeon, B. J. 2014 On the structure and origin of pressure fluctuations in wall turbulence: predictions based on the resolvent analysis. J. Fluid Mech. 751, 38–70.Google Scholar
Lumley, J. L. 1967 The structure of inhomogeneous turbulence. In Atmospheric Turbulence and Wave Propagation, pp. 166–178. Nauka.Google Scholar
Nagarajan, S., Lele, S. K. & Ferziger, J. H. 2003 A robust high-order compact method for large eddy simulation. J. Comput. Phys. 191 (2), 392–419.Google Scholar
Nair, V. & Hinton, G. E. 2010 Rectified linear units improve restricted Boltzmann machines. In Proceedings of the 27th International Conference on International Conference on Machine Learning, pp. 807–814. Omnipress.Google Scholar
Navarro, L. C., Navarro, A. K. W., Rocha, A. & Dahab, R. 2018 Connecting the dots: Toward accountable machine-learning printer attribution methods. J. Visual Commun. Image Represent. 53, 257–272.Google Scholar
Noack, B. R., Afanasiev, K., Morzynski, M., Tadmor, G. & Thiele, F. 2003 A hierarchy of low-dimensional models for the transient and post-transient cylinder wake. J. Fluid Mech. 497, 335–363.Google Scholar
Ol, M. V., Bernal, L., Kang, C. & Shyy, W. 2009 Shallow and deep dynamic stall for flapping low Reynolds number airfoils. Exp. Fluids 46 (5), 883–901.Google Scholar
Olson, B. J.2012 Large-eddy simulation of multi-material mixing and over-expanded nozzle flow. PhD thesis, Stanford University.Google Scholar
Östh, J., Noack, B. R., Krajnovi, S., Barros, D. & Borée, J. 2014 On the need for a nonlinear subscale turbulence term in POD models as exemplified for a high-Reynolds-number flow over an Ahmed body. J. Fluid Mech. 747, 518–544.Google Scholar
Pan, S. & Duraisamy, K.2018 Long-time predictive modeling of nonlinear dynamical systems using neural networks. arXiv:1805.12547.Google Scholar
Protas, B., Noack, B. R. & Östh, J. 2015 Optimal nonlinear eddy viscosity in Galerkin models of turbulent flows. J. Fluid Mech. 766, 337–367.Google Scholar
Ribeiro, J. H. M. & Wolf, W. R. 2017 Identification of coherent structures in the flow past a NACA0012 airfoil via proper orthogonal decomposition. Phys. Fluids 29 (8), 085104.Google Scholar
Rockwood, M. P., Taira, K. & Green, M. A. 2016 Detecting vortex formation and shedding in cylinder wakes using Lagrangian coherent structures. AIAA J. 55, 15–23.Google Scholar
Rowley, C. W., Colonius, T. & Murray, R. M. 2004 Model reduction for compressible flows using POD and Galerkin projection. Physica D: Nonlinear Phenomena 189 (12), 115–129.Google Scholar
Rudy, S. H., Brunton, S. L., Proctor, J. L. & Kutz, N. J. 2017 Data-driven discovery of partial differential equations. Sci. Adv. 3 (4), e1602614.Google Scholar
Rudy, S. H., Kutz, J. N. & Brunton, S. L.2018 Deep learning of dynamics and signal-noise decomposition with time-stepping constraints. arXiv:1808.02578.Google Scholar
San, O. & Maulik, R. 2018 Extreme learning machine for reduced order modeling of turbulent geophysical flows. Phys. Rev. E 97, 042322.Google Scholar
Schmid, P. J. 2010 Dynamic mode decomposition of numerical and experimental data. J. Fluid Mech. 656, 5–28.Google Scholar
Schwarz, G. 1978 Estimating the dimension of a model. Ann. Stat. 6, 461–464.Google Scholar
Sharma, A. S., Moarref, R., McKeon, B. J., Park, J. S., Graham, M. D. & Willis, A. P. 2016 Low-dimensional representations of exact coherent states of the Navier–Stokes equations from the resolvent model of wall turbulence. Phys. Rev. E 93, 021102.Google Scholar
Sieber, M., Paschereit, C. O. & Oberleithner, K. 2016 Spectral proper orthogonal decomposition. J. Fluid Mech. 792, 798–828.Google Scholar
Sirovich, L. 1986 Turbulence and the dynamics of coherent structures. Part I. Coherent structures. Q. Appl. Maths 45, 561–571.Google Scholar
Snoek, J., Larochelle, H. & Adams, R. P. 2012 Practical Bayesian optimization of machine learning algorithms. In Advances in Neural Information Processing Systems 25, pp. 2951–2959. Curran Associates.Google Scholar
Ströfer, C. M., Wu, J. L., Xiao, H. & Paterson, E. 2019 Data-driven, physics based feature extraction from fluid flow fields using convolutional neural networks. Commun. Comput. Phys. 25, 625–650.Google Scholar
Taira, K., Brunton, S. L., Dawson, S. T. M., Rowley, C. W., Colonius, T., McKeon, B. J., Schmidt, O. T., Gordeyev, S., Theofilis, V. & Ukeiley, L. S. 2017 Modal analysis of fluid flows: an overview. AIAA J. 55 (12), 4013–4041.Google Scholar
Towne, A., Schmidt, O. T. & Colonius, T. 2018 Spectral proper orthogonal decomposition and its relationship to dynamic mode decomposition and resolvent analysis. J. Fluid Mech. 847, 821–867.Google Scholar
Tu, J. H., Rowley, C. W., Luchtenburg, D. M., Brunton, S. L. & Kutz, N. J. 2014 On dynamic mode decomposition: theory and applications. J. Comput. Dyn. 1, 391–421.Google Scholar
Visbal, M. R. 2011 Numerical investigation of deep dynamic stall of a plunging airfoil. AIAA J. 49, 2152–2170.Google Scholar
Vlachas, P. R., Byeon, W., Wan, Z. Y., Sapsis, T. P. & Koumoutsakos, P. 2018 Data-driven forecasting of high-dimensional chaotic systems with long short-term memory networks. Proc. R. Soc. Lond. A 474 (2213).Google Scholar
Wan, Z. Y., Vlachas, P., Koumoutsakos, P. & Sapsis, T. 2018 Data-assisted reduced-order modeling of extreme events in complex dynamical systems. PLOS ONE 13 (5), 1–22.Google Scholar
Wang, J. X., Wu, J. L. & Xiao, H. 2017 Physics-informed machine learning approach for reconstructing Reynolds stress modeling discrepancies based on DNS data. Phys. Rev. Fluids 2, 034603.Google Scholar
Wolf, W. R.2011 Airfoil aeroacoustics: LES and acoustic analogy predictions. PhD thesis, Stanford University.Google Scholar
Wolf, W. R., Azevedo, J. L. F. & Lele, S. K. 2012a Convective effects and the role of quadrupole sources for aerofoil aeroacoustics. J. Fluid Mech. 708, 502–538.Google Scholar
Wolf, W. R., Lele, S. K., Jothiprasard, G. & Cheung, L.2012 Investigation of noise generated by a DU96 airfoil. AIAA Paper 2012-2055.Google Scholar
Xavier, G. & Yoshua, B. 2010 Understanding the difficulty of training deep feedforward neural networks. In Proceedings of the Thirteenth International Conference on Artificial Intelligence and Statistics (ed. Teh, Y. W. & Titterington, M.).Google Scholar
Zimmermann, R. & Willcox, K. 2016 An accelerated greedy missing point estimation procedure. J. Sci. Comput. 38, 2827–2850.Google Scholar

Lui and Wolf supplementary movie 1

Movie showing x-momentum for full and reduced order models (cylinder flow)

Download Lui and Wolf supplementary movie 1(Video)
Video 9.3 MB

Lui and Wolf supplementary movie 2

Movie showing Q-criterion colored by pressure for full and reduced order models (3D flow of plunging airfoil)

Download Lui and Wolf supplementary movie 2(Video)
Video 49.4 MB

Lui and Wolf supplementary movie 3

Movie showing x-momentum for full and reduced order models (spanwise-averaged flow of plunging airfoil)

Download Lui and Wolf supplementary movie 3(Video)
Video 12.1 MB